DERIVATIVES OF ABSOLUTE VALUE FUNCTIONS WORKSHEET

Find the derivative of each of the following absolute value functions with respect to x.

1) |2x + 1|

2) |x³ +  1|

3) |x|³

4) |2x - 5|

5) (x - 2)² + |x - 2|

6) 3|5x + 7|

7) |sinx|

8) |cosx|

9) |tanx|

10) |sinx + cosx|

1. Answer :

|2x + 1|' = [(2x + 1)/|2x + 1|](2x + 1)'

= [(2x+1)/|2x+1|](2)

= 2(2x+1)/|2x+1|

2. Answer :

|x³ + 1|' = [(x³ + 1)/|x³ + 1|](x³ + 1)'

= [(x³ + 1)/|x³ + 1|](3x²)

= 3x²(x³ + 1)/|x³ + 1|

3. Answer :

In the given function |x|³, using chain rule, first we have to find derivative for the exponent 3 and then for |x|.

(|x|³)' = {3|x|²}[x/|x|](x)'

= {3|x|²}[x/|x|](1)

= 3x|x|

4. Answer :

|2x - 5|' = [(2x - 5)/|2x - 5|](2x-5)'

= [(2x - 5)/|2x - 5|](2)

= 2(2x - 5)/|2x - 5|

5. Answer :

{(x - 2)² + |x - 2|}' = [(x - 2)²]' + |x - 2|'

= 2(x - 2) + [(x - 2)/|x - 2|](x - 2)'

= 2(x - 2) + [(x - 2)/|x - 2|](1)

= 2(x - 2) + (x - 2)/|x - 2|

6. Answer :

[3|5x+7|]' = 3[(5x + 7)/|5x + 7|](5x+7)'

 = 3[(5x + 7)/|5x + 7|](5)

= 15(5x + 1)/|5x + 7|

7. Answer :

|sinx|' = [sinx/|sinx|](sinx)'

= [sinx/|sinx|]cosx

= (sinx ⋅ cosx)/|sinx|

8. Answer :

|cosx|' = [cosx/|cosx|](cosx)'

= [cosx/|cosx|](-sinx)

= -(sinx ⋅ cosx)/|cosx|

9. Answer :

|tanx|' = [tanx/|tanx|](tanx)'

= [tanx/|tanx|]sec²x

=  (sec²x ⋅ tanx)/|tanx|

10. Answer :

|sinx + cosx|' = [(sinx + cosx)/|sinx + cosx|](sinx + cosx)'

= [(cosx + sinx)/|sinx + cosx|](cosx - sinx)

= (cos²x - sin²x)/|sinx + cosx|

= cos2x/|sinx + cosx|

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